Bardone, Lorenzo (2019) Ising model on deterministic and random graphs: some examples. [Laurea triennale]
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The Ising model is one of the simplest mathematical settings in which it can be studied how, from a local correlation of the particles, overall order can arise. This work provides an overview on this model, with particular attention to examples, both in deterministic and random graphs. The structure of the thesis is the following: after a brief introduction on the most popular random graphs models, we present and define the Ising model itself, then there is the largest part of this work, in which we analyze some of its applications. Among the deterministic examples it is worth to mention our presentation of Curie-Weiss model, in which we prove the presence of phase transition and also limit results on the pressure and total spin. The most of attention is dedicated to the last example, the Ising model application on (a variant of) the Erdös-Renyi random graph. After some introductory work, we study and make a comparison between the high energy region and the temperature zero setting. In this way we reach the conclusion that the thermodynamic limit of the pressure per particle is not analytic.
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